The present invention relates generally to micromechanical resonators, and more particularly, to process-variation-insensitive, temperature-stable micromechanical resonators.
Micromechanical resonators are strong candidates for integrated microelectronic solutions requiring a stable frequency reference. In particular, they have strong potential for replacement of large off-chip quartz resonators. Micromechanical resonators, fabricated in a manner similar to microelectronic devices, are subject to processing variations. As the size of substrates (wafers) increase, the variation of critical dimension (CD) variation typically increases. For this reason, planar resonator dimensions, and subsequently, their natural frequency, will vary on the substrate. Disclosed is a design for manufacturability (DFM) scheme to reduce the frequency divergence caused by critical dimension variations to target two potential applications: real time clocks (RTC) and high frequency reference oscillators.
Another performance metric in frequency references is temperature stability. In most materials, the acoustic velocity να is dependent on temperature. The natural frequency of a structure is directly proportional to the να and inversely proportional to dimension L:
      f    =                            β          i                L            ⁢              v        a              ,          ⁢            v      a        =                  C        ρ            where βi is a dimensionless parameter. In the most simplified form, the acoustic velocity is the square root of the stiffness coefficient C over density ρ.
Temperature stability of the resonator is ensured when the fractional temperature gradient of acoustic velocity γνα is equal to the linear thermal expansion coefficient α since
            1      f        ⁢                  ⅆ        f                    ⅆ        t              =                              -                      1            L                          ⁢                              ⅆ            L                                ⅆ            T                              +                        1                      v            a                          ⁢                              ⅆ                          v              a                                            ⅆ            T                                =                  -        α            +                        γ          va                .            
In general, materials have a positive α. With the rare exception of quartz and silicon dioxide, materials have a negative γνα. Therefore, resonators made of semiconductor materials exclusively will always have an undesirable temperature sensitivity. It would be desirable to reduce the inherent temperature dependence using a material with positive γνα. It would also be desirable to have process-variation-insensitive, temperature-stable micromechanical resonators.